#6558588

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,8)(4,6)(5,7)(17,18,23,27)(19,24,29,39)(20,31,34,43)(21,26,22,37)(25,36,44,33)(28,45,32,48)(30,40,35,41)(38,46,42,47);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,32)(21,35)(22,30)(23,29)(25,42)(26,41)(27,39)(28,34)(31,48)(33,46)(36,47)(37,40)(38,44)(43,45);(10,13)(14,16)(17,20,21,36,23,34,22,33)(18,25,37,43,27,44,26,31)(19,28,35,46,29,32,30,47)(24,38,40,48,39,42,41,45);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,29)(24,39)(28,32)(30,35)(38,42)(40,41)(45,48)(46,47);(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,30,29,35)(20,33,34,36)(24,40,39,41)(25,43,44,31)(28,46,32,47)(38,45,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^c=ei,e^d=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6465784);